\( {\mathcal{Z}}^{\ast } \)-Semilocal Modules and the Proper Class \( \mathrm{\mathcal{R}}\mathcal{S} \)

  • E. TürkmenEmail author

Over an arbitrary ring, a module M is said to be \( {\mathcal{Z}}^{\ast } \)-semilocal if every submodule U of M has a \( {\mathcal{Z}}^{\ast } \) -supplement V in M, i.e., M = U + V and \( U\cap \kern0.5em V\subseteq {\mathcal{Z}}^{\ast }(V), \) where \( {\mathcal{Z}}^{\ast }(V)=\left\{m\in \left.V\right| Rm\kern0.5em \mathrm{is}\kern0.5em \mathrm{a}\kern0.5em \mathrm{small}\kern0.5em \mathrm{module}\right\} \) is the Rad-small submodule. We study basic properties of these modules regarded as a proper generalization of semilocal modules. In particular, we show that the class of \( {\mathcal{Z}}^{\ast } \) -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring R is \( {\mathcal{Z}}^{\ast } \) -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class \( \mathrm{\mathcal{R}}\mathcal{S} \) of all short exact sequences \( \mathbbm{E}:0\to M\overset{\psi }{\to }N\overset{\phi }{\to }K\to 0 \) such that Im(ψ) has a \( {\mathcal{Z}}^{\ast } \) -semilocal in N is a proper class over left hereditary rings. We also study some homological objects of the proper class \( \mathrm{\mathcal{R}}\mathcal{S} \).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Alizade, E. Büyükaşık, and Y. Durgun, “Small supplements, weak supplements, and proper classes,” Hacet. J. Math. Stat., 45, No. 3, 449–461 (2016).MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Alizade, “Global dimension of some proper class,” Usp. Mat. Nauk, 1, 181–182 (1985).Google Scholar
  3. 3.
    R. Alizade, Y. M. Demirci, Y. Durgun, and D. Pusat, “The proper class generated by weak supplements,” Comm. Algebra, 42, 56–72 (2014).MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. A. Buchsbaum, “A note on homology in categories,” Ann. Math., 69, 66–74 (1959).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. I. Generalov, The 𝜔-Cohigh Purity in Categories of Modules, Plenum, New York (1983).zbMATHGoogle Scholar
  6. 6.
    F. Kasch, Modules and Rings, Academic Press, London (1982).zbMATHGoogle Scholar
  7. 7.
    W. W. Leonard, “Small modules,” Proc. Amer. Math. Soc., 17, 527–531 (1966).MathSciNetCrossRefGoogle Scholar
  8. 8.
    C. Lomp, “On semilocal modules and rings,” Comm. Algebra, 27, No. 4, 1921–1935 (1999).MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Mac Lane, Homology, Academic Press, New York (1963).CrossRefGoogle Scholar
  10. 10.
    E. Mermut, Homological Approach to Complements and Supplements, PhD Thesis (2004).Google Scholar
  11. 11.
    A. P. Misina and L. A. Skornjakov, Abelian Groups and Modules, Amer. Math. Soc. (1960).Google Scholar
  12. 12.
    R. J. Nunke, “Purity and subfunctors of the identity,” in: Topics in Abelian Groups: Proc. Symp., New Mexico State University, 3 (1963), pp. 121–171.Google Scholar
  13. 13.
    A. Ç. Özcan, “Modules with small cyclic submodules in their injective hulls,” Comm. Algebra, 30, No. 4, 1575–1589 (2002).MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Wisbauer, Foundations of Modules and Rings, Gordon & Breach (1991).Google Scholar
  15. 15.
    H. Zöschinger, “Komplementierte moduln über Dedekindringen,” J. Algebra, 29, 42–56 (1974).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Amasya UniversityAmasyaTurkey

Personalised recommendations